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Rational surfaces of Kodaira type IV. (English) Zbl 1177.14077

Summary: We study the geometry of a rational surface of Kodaira type IV by giving the nature of its integral curves of self-intersection less than zero, in particular, we show that they are smooth and rational. Hence, under a reasonable assumption, we prove the finite generation of its monoid of effective divisor classes and in almost all cases its anticanonical complete linear system is of projective dimension zero and of self-intersection strictly negative. Furthermore, we show that if this condition is not fulfilled, the monoid may fail to be finitely generated.

MSC:

14J26 Rational and ruled surfaces
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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